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On the SL(2, C)-representation rings of free abelian groups

Published online by Cambridge University Press:  28 May 2018

TAKAO SATOH
TAKAO SATOH*
Affiliation:
Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo, 162-8601, Japan. e-mail: takao@rs.tus.ac.jp
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Abstract

In this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.

Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 2 , September 2019 , pp. 229 - 247
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

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